# Linear Algebra for Data Science

### A mini-course in Linear Algebra for Data Science

This online workshop reviews topics in linear algebra to help students prepare for MATH 677 Mathematical Foundations for Data Science in the Masters of Science in Data Science at Texas A&M University.

For more information on the M.S. in Data Science, please consult the program pages at the Texas A&M Institute of Data Science for further information.

This course is a collaboration between the Math Learning Center and the Texas A&M Institute of Data Science.

Resources

This mini-course references and uses exercises from the following textbooks:

*Linear Algebra with Applications*by W. Keith Nicholson; available as an open educational resource*Introduction to Linear Algebra*by R. Fioresi and M. Morigi; available through TAMU Library as an e-book

### Format

- Each session below contains a video covering the listed topics.
- Each video has homework problems you should attempt after watching the video.

## Course Content

View Session 1

- Definition of vectors
- Systems of linear equations and their solution sets
- Homogeneous linear systems

- Augmented matrix for a systems of equations
- Elementary row operations and row echelon form
- Solving systems of equations using Gaussian Elimination
- Using Gauss-Jordan Reduction to put an augmented matrix in reduced row echelon form

- Symmetric Matrices, the transpose, and the trace
- Linear combinations of matrices
- Matrix multiplication and its properties
- The inverse of a matrix and its properties

- Properties of elementary matrices
- Finding the inverse of a matrix
- Properties of invertible matrices
- The determinant of a matrix

- Calculating the determinant of a matrix
- Properties of the determinant
- How row operations change the determinant
- The determinant of invertible and noninvertible matrices

- Cramer's Rule
- The adjoint and the inverse of a matrix
- The definition and properties of vector spaces
- Subspaces of a vector space

- The span of a subset and its properties
- The definition and properties of linear independence
- Basis for a vector space

- Determining if a set is a basis for a vector space
- Properties of a basis for a vector space
- The dimension of a vector space
- The column, row, and null spaces of a matrix
- Rank and nullity of a matrix
- Coordinates of a vector space

- Using a basis of a vector space to create an isomorphism to \(\mathbb{R}^n\)
- Finding a basis of a subspace from a spanning set
- Change of basis

- Rank-Nullity Theorem
- Linear transformations
- Kernel and range of linear transformations
- Definition of an isomorphism

- Properties of linear transformations including injective, surjective, and bijective
- Matrix representations of linear transformations between vector spaces
- The relationship between the matrix for a linear transformation and the range and kernel of the linear transformation
- Rank-Nullity Theorem (or Dimension Theorem) for linear transformations

- Examples of linear transformations and the application of the Rank-Nullity Theorem
- Relationship between transition matrices and linear transformations
- Using transition matrices to change the matrix representation of a linear representation to use a different basis of the vector space
- Similarity of matrices
- Eigenvalues and eigenvectors

- Properties of eigenvalues and eigenvectors
- Definition of the eigenspace of an eigenvalue
- Characteristic polynomial of a matrix
- Algebraic and geometric multiplicity of an eigenvalue
- Determining if a matrix is diagonalizable
- Vector spaces with norms